Thanks for all the comments. Both of our previous trips taught hubby and me that we splash too much to contain the water (even when we turned off water between soaping up and rinsing off). We just felt bad making a mess, not to mention that we hate stepping onto wet floors too. And sometimes the hotel staff is not too pleased to provide extra towels.
Shower shoes are an option I hadn't thought about, so I'm going to look for some to try out. If they don't work, then my trusty bath mat goes with me. Since it rolls up so tightly, it doesn't take up a lot of room in my suitcase. I'd rather have it than risk a nasty fall.
In a pinch, I've also used a hand towel in place of a bath mat. It's better than nothing. Again, sometimes extras are hard to get from the hotel staff.
I guess I'll stop worrying about soaking floors and just enjoy my vacation! LOL!